Show that if available carpentry hours remain between 60 and 100, the current basis remains optimal. If between 60 and 100 carpentry hours are available, would Giapetto still produce 20 soldiers and 60 trains?

Show that if the contribution to profit for trains is between $1.50 and$3, the current basis remains optimal. If the contribution to profit for trains is $2.50, then what would be the new optimal solution?

Show that if the weekly demand for soldiers is at least 20, the current basis remains optimal, and Giapetto should still produce 20 soldiers and 60 trains.

Suppose we are solving a maximization problem and the variable xr is about to leave the basis. a. What is the coefficient of xt. in the current row 0? b. Show that after the current pivot is performed, the coefficient of xr in row 0 cannot be less than zero. c. Explain why a variable that has left the basis on a given pivot cannot re-enter the basis on the next pivot.

Radioco manufactures two types of radios. The only scarce resource needed to produce radios is labor. The company now has two laborers. Laborer 1 is willing to work as many as 40 hours per week and is paid $5 per hour. Laborer 2 is willing to work up to 50 hours per week and is paid$6 per hour. The price as well as the resources required to build each type of radio are given in Table 1. Letting xi be the number of type i radios produced each week, show that Radioco should solve the following LP:

$$\begin{array}{c}{\max z=3 x_{1}+2 x_{2}} \\ {\text { s.t. } \quad x_{1}+2 x_{2} \leq 40} \\ {2 x_{1}+x_{2} \leq 50} \\ {x_{1}, x_{2} \geq 0}\end{array}$$

a. For what values of the price of a Type 1 radio would the current basis remain optimal? b. For what values of the price of a Type 2 radio would the current basis remain optimal? c. If laborer 1 were willing to work only 30 hours per week, would the current basis remain optimal? Find the new optimal solution to the LP. d. If laborer 2 were willing to work up to 60 hours per week, would the current basis remain optimal? Find the new optimal solution to the LP. e. Find the shadow price of each constraint. Table 1:

$$\begin{matrix} \text{ } & \text{Radio 1} & \text{ } & \text{Radio 2}\\ \text{Price (\$)} & \text{Resource Required} & \text{Price (\$)} & \text{Resource Required}\\ \text{25} & \text{Laborer 1: 1 hour} & \text{22} & \text{Laborer 1: 2 hours.}\\ \text{ } & \text{1 hours} & \text{ } & \text{Laborer 2: 2 hours}\\ \text{ } & \text{Raw material cost: \$5} & \text{ } & \text{Raw material cost: \$4}\\ \end{matrix}$$

Radioco manufactures two types of radios. The only scarce resource that is needed to produce radios is labor. At present, the company has two laborers. Laborer 1 is willing to work up to 40 hours per week and is paid $5 per hour. Laborer 2 will work up to 50 hours per week for$6 per hour. The price as well as the resources required to build each type of radio are given in Table 1. Letting xi be the number of Type i radios produced each week, Radioco should solve the following LP:

$$\begin{array}{c}{\max z=3 x_{1}+2 x_{2}} \\ {\text { s.t. } \quad x_{1}+2 x_{2} \leq 40} \\ {2 x_{1}+x_{2} \leq 50} \\ {x_{1}, x_{2} \geq 0}\end{array}$$

a. For what values of the price of a Type 1 radio would the current basis remain optimal? b. For what values of the price of a Type 2 radio would the current basis remain optimal? TABLE 1:

$$\begin{matrix} \text{Radio 1} & \text{ } & \text{Radio 2} & \text{ }\\ \text{Price (\$)} & \text{Resource Required} & \text{Price (\$)} & \text{Resource Required}\\ \text{25} & \text{Laborer 1: 1 hour} & \text{22} & \text{Laborer 1: 2 hours}\\ \text{ } & \text{Laborer 2: 2 hours} & \text{ } & \text{Laborer 2: 2 hours}\\ \text{ } & \text{Raw material cost: \$5} & \text{ } & \text{Raw material cost: \$4}\\ \end{matrix}$$

c. If laborer 1 were willing to work only 30 hours per week, then would the current basis remain optimal? Find the new optimal solution to the LP. d. If laborer 2 were willing to work up to 60 hours per week, then would the current basis remain optimal? Find the new optimal solution to the LP. e. Find the shadow price of each constraint.

There are four teachers in the Faber College Business School. Each semester, 200 students take each of the following courses: marketing, finance, production, and statistics. The “effectiveness” of each teacher in teaching each class is given in Table 61. Each teacher can teach a total of 200 students during the semester. The dean has set a goal of obtaining an average teaching effectiveness level of about 6 in each course. Deviations from this goal in any course are considered equally important. Formulate a goal programming model that can be used to determine the semester’s teaching assignments. TABLE 61:

$$\begin{matrix} \text{Teacher} & \text{Marketing} & \text{Finance} & \text{Production} & \text{Statistics}\\ \text{1} & \text{7} & \text{5} & \text{8} & \text{2}\\ \text{2} & \text{7} & \text{8} & \text{9} & \text{4}\\\text{3} & \text{3} & \text{5} & \text{7} & \text{9}\\ \text{4} & \text{5} & \text{5} & \text{6} & \text{7}\\ \end{matrix}$$

Identify the letter of the word whose meaning is closest to that of the first word. effigies: (a) speeches, (b) insults, (c) dummies

Reflect about what you have learned from today's lesson as you answer the following question in your Learning Log. Title this entry "Representing Expressions on an Expression Mat" and include today's date . Using an Expression Mat, find two different ways to represent $x -1- (2x - 3)$ . Sketch the different representations and write a few sentences to describe the differences in the ways you built each representation.

Complete each statement. ? m = 3.01 km

"Rocket man" has a propulsion unit strapped to his back. He starts from rest on the ground, fires the unit, and is propelled straight upward. At a height of 16 m, his speed is 5.0 m/s. His mass, including the propulsion unit, has the approximately constant value of 136 kg. Find the work done by the force generated by the propulsion unit.

Solve each equation below. $\frac{5 x+4}{3}=\frac{2 x+1}{5}$

Determine whether the given value is a solution of the equation.

$$\frac{36}{x}=9 ; x=4$$

Find the volume of the solid E whose boundaries are given in rectangular coordinates. E is bounded by the circular cone $z=\sqrt{x^{2}+y^{2}}$ and z=1.